Algebraically recurrent random walks on groups
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Benjamini, Itai, Hilary Finucane, and Romain Tessera. “Algebraically
Recurrent Random Walks on Groups.” Electronic Communications in
Probability 18, no. 0 (January 3, 2013).
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http://dx.doi.org/10.1214/ECP.v18-2519
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Institute of Mathematical Statistics
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Final published version
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http://hdl.handle.net/1721.1/89829
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ISSN: 1083-589X in PROBABILITY
Algebraically recurrent random walks on groups
Itai Benjamini
∗Hilary Finucane
†Romain Tessera
‡Abstract
Initial steps are presented towards understanding which finitely generated groups are almost surely generated as a semigroup by the path of a random walk on the group.
Keywords: random walks on groups, recurrence/transience, semi-groups. AMS MSC 2010: 05C81.
Submitted to ECP on December 25, 2012, final version accepted on April 12, 2013.
1
Introduction
LetGbe a countable group,µbe a probability measure onG,ζi∼ µbe i.i.d., and let
Xn= ζ1ζ2· · · ζn. Then we call(X1, X2, . . .)aµ-random walk onG. Since Furstenberg [1]
qualitative properties of random walks on groups were used to study and classify natu-ral properties of groupsGor pairs(G, µ)such as the Liouville property and amenability; see e.g. [3]. In this work we define a new group property based on random walks, which we call algebraic recurrence, and we present some initial steps towards understanding which groups are algebraically recurrent.
Definition 1.1. Let(X1, X2, . . .)be aµ-random walk onG, and letSndenote the
semi-group generated by{Xn, Xn+1, . . .}. We say(G, µ)is algebraically recurrent (AR) if for
alln,Sn = Galmost surely, and we callGAR if(G, µ)is AR for all symmetric measures
µwithhsupp(µ)i = G.
Most classical properties of random walks on groups, such as recurrence/transience, Liouville property, etc. can be abstracted from the context of groups. By contrast, the definition of algebraic recurrence requires at least some binary operation on the state set of the random walk.
The use of a semigroup rather than a subgroup in the definition may seem unnat-ural, but in fact the property is trivial if defined in terms of subgroups rather than semigroups. To see this, letGn denote the group generated by{Xn, Xn+1, . . .}and
sup-posehsupp(µ)i = G. Then for eachg ∈ supp(µ),Pr(g /∈ {ζn+1, ζn+2, . . .}) = 0, but for all
i > n, ζi = (Xi−1)−1Xi ∈ Gn. Thus,supp(µ) ⊂ Gn, and so Gn = G almost surely. This
argument in fact proves the more general fact:
Lemma 1.2. IfXi−1 ∈ Snalmost surely for alli ≥ n, thenSn = Galmost surely.
∗Weizmann Institute, Israel †MIT, USA
Algebraically recurrent random walks on groups
In this note we show that the class of AR groups is nontrivial: i.e. there exist AR and non-AR groups. For example, we prove that nilpotent finitely generated groups are AR, while free groups with more than 4 generators are not AR. We also prove that Liouville random walks on polycyclic groups are AR. By [2], this includes symmetric random walks with a first finite moment. We do not know if this fact extends to any symmetric random walk on a polycyclic group.
We have to admit our frustration at not being able to establish that the standard random walk on the free group with two (or 3) generators is not AR. It turns out that Theorem 3.1 is by far the trickiest result of this paper and we would be curious to see another –more geometric– proof of this fact.
In view of the fact that a non-centered random walk on Z is trivially not AR, the assumption that our random walks are symmetric seems reasonable. However, one would legitimately be tempted to extend the results of this note to centered random walks: namely random walks whose projection to any cyclic quotient is centered. We leave this aspect of the question to future developments.
The paper is organized as follows: in Section 2, we give examples of AR groups, then Section 3 deals with the case of free groups. Finally Section 4 is dedicated to open questions.
2
Examples of AR groups
2.1 Torsion groups and lamplightersRecall thatGis a torsion group if every element ofGhas finite order. By Lemma 1.2, any torsion group is AR, since in a torsion group,Xi−1= Xm
i for somem, and soX −1 i is
in any semigroup that includesXi. In fact, the following much stronger result holds.
Theorem 2.1. SupposeH / Gis a torsion group. ThenG/His AR if and only ifGis AR. Proof. SupposeG/His AR, and letπdenote the projection fromGtoG/H. Let(X1, X2, . . .)
be aµ-random walk onGwith corresponding semigroupSn, and letS¯nbe the semigroup
ofG/Hcorresponding to the projected random walk(π(X1), π(X2), . . .). By Lemma 1.2,
to show thatGis AR, it suffices to showXi−1∈ Snfor alli ≥ n.
For any X and Y in G with π(X) = π(Y )−1, we have XY ∈ H, so there is an exponentksuch that(XY )k = e, and thusX−1 = Y (XY )k−1. So if there is aY
i inSn
withπ(Yi) = π(Xi)−1, then we also haveXi−1∈ Sn.
By the algebraic recurrence ofG/H, we haveG/H = ¯Sn, and in particular,π(Xi)−1∈
¯
Sn. But sinceS¯n = π(Sn), this means that there is someYi inSn withπ(Yi) = π(Xi)−1.
SoGis AR.
Conversely, ifGis AR, then any random walk onG/Hcan be lifted to a random walk onG; the corresponding semigroup is all ofG, and so projects to all ofG/H, showing thatG/His AR.
The lamplighter group of a groupG, denotedLL(G), is the wreath productZ/2Z o G. An element ofLL(G)is written(x, f ), wherex ∈ Gandf : G → Z/2Z, and(x, f )(y, g) = (z, h), wherexy = zandh(a) = f (a)g(ax−1).
Lamplighters often give examples of somewhat exotic behavior; for example,LL(Z) has exponential growth but is Liouville and LL(Z3)is amenable but non-Liouville [3].
It follows directly from Theorem 2.1 that lamplighters do not exhibit any unusual be-havior in the case of algebraic recurrence: in fact, the algebraic recurrence ofLL(G)
corresponds exactly to the algebraic recurrence ofG.
ECP 18 (2013), paper 28.
Corollary 2.2. LL(G)is AR if and only ifGis AR.
Proof. The position functionpos(x, f ) = xis a surjective homorphism fromLL(G)toG, and the kernel ofposis a torsion group with exponent two.
2.2 Finitely generated abelian groups
Lemma 1.2 can also be used to show thatZ is AR. Indeed, by the symmetry of µ, almost surely for allnthere will bey+, y− ∈ S
n withy+ > 0andy−< 0. Then for each
i ≥ n, if Xi > 0we can writeXiy− + (−y−− 1)Xi = −Xi. But the LHS is in Sn, so
−Xi∈ Sn. Similarly, ifXi < 0we have(y+− 1)Xi+ −Xiy+= −Xi, and again the LHS
is inSnso−Xi∈ Sn. Using Lemma 1.2, this suffices to show thatZis AR.
It is not much more difficult to see that(Zd, µ
d)is AR, whenµdis uniform over the
standard generating set. Indeed, after finitely many steps, the random walk will have visited d linearly independent points, generating the intersection of a full-dimension lattice with a cone. Eventually, the random walk will visit a pointxthat is in the opposite cone, and by adding arbitrarily large multiples of x, the entire lattice is in Sn. Since
there are only finitely many cosets of the lattice, the random walk eventually visits each coset, showing that all ofZdis inS
n.
However, this proof does not extend to arbitrary symmetric generating measures on
Zd. For example, in
Z2, µcould have a very heavy tail along the line x = yand a very
small weight along the linex = −y, so that there are cones that the random walk has non-zero probability never to intersect. So for the general case, a more subtle proof is needed.
Clearly,(G, µ)is AR if and only if the trace of aµ-random walk onGis almost surely not contained in any maximal subsemigroup of G. In the case of Zd, these maximal
subsemigroups are easy to describe.
Lemma 2.3. Every proper subsemigroup ofZdis contained either in a proper subgroup
ofZd or in a half-space of
Zd.
Proof. LetSbe a subgroup ofZd. If0is not in the convex hull ofS, then there is a
halfs-pace containingS. Otherwise, by Caratheodory’s theorem, there are pointsx1, . . . , xd+1
and positive numberst1, . . . , td+1 such that P tixi = 0 andx1, . . . , xd are linearly
inde-pendent. Thus,xd+1is written as a linear combination ofx1, . . . xd, using only negative
coefficients. This allows us to generate arbitrary linear combinations ofx1, . . . , xd, so
the groupH generated by x1, . . . , xd is contained inS. LetS¯ denote the projection of
S toZd/H. Because
Zd/H is torsion, S¯ is a subgroup. IfS = Z¯ d/H, then
S = Zd.
Otherwise,Sis contained in a proper subgroup ofZd.
Definition 2.4. LetGbe a countable group. Denote byS(G)the set of subsemi-groups ofG. Note that S(G) is a compact space for the product topology (hence a standard Borel space). The inverse of some semigroupH is the semigroup consisting of inverses of elements ofH. Let(Hn)be a decreasing sequence ofS(G)-valued random variables.
We shall say that(Hn)is
• non-degenerate ifHngeneratesGas a subgroup for allnalmost surely;
• Liouville if the tailσ-algebra is trivial;
• symmetric if for everyn, and every Borel subsetΩ ⊂ S(G), the events{Hn ⊂ Ω}
and{H−1
n ⊂ Ω}are equiprobable.
Clearly, if (Xn) is a µ-random walk with µ symmetric and non-degenerate, then the
Algebraically recurrent random walks on groups
Theorem 2.5. Zd is AR for alld ≥ 1. More generally, every non-degenerate Liouville
symmetric decreasing sequence of random semigroups(Hn)ofZdis such thatHn= Zd
a.s. for alln.
Proof. The proof immediately follows from Lemma 2.3 together with the following lemma.
Lemma 2.6. Let G be a countable subgroup of Rd and let (H
n) be some Liouville
symmetric decreasing sequence of random semigroups of G, such that for all n, Hn
generates Rd as a vector space. Then a.s. H
n does not eventually get trapped in a
closed half-space ofRd.
Proof. We will prove the lemma by induction ond, the cased = 0being trivial. LetAn
the closure of the radial projection ofHn to the sphereSd−1inRd. By compactness of
the sphere, the intersection of theAn’s is a non-empty closed subsetA ⊂ Sn−1.
Claim:Ais a deterministic set; i.e. there exists a setT ⊂ Sn−1such thatPr(A = T ) = 1.
Moreover,A = −Aalmost surely.
The symmetry ofAfollows from the symmetry ofHn. To show thatAis deterministic,
we use the Liouville property of(Hn): A depending only on the tail ofHn, any event
depending only onAhas probability 0 or 1. But the only probability measure on closed sets that satisfies this property is the Dirac measure; in particular, there is one closed setT such thatP (A = T ) = 1.
By the Claim, there is a pair of pointsx, −xinSd−1that are almost surely contained
inA. Thus,Hn is almost surely not contained in any halfspace that does not contain
xand −x. Letπ denote the projection onto the hyperplane orthogonal tox. IfHn is
eventually contained in halfspace containingxand−x, thenπ(Hn)must be contained
in a halfspace ofRd−1. But the projectionπ(H
n)generatesRd−1as a vector space (and
is obviously Liouville and symmetric). Hence the lemma follows by induction on the dimension.
Remark 2.7. The level of generality of Theorem 2.5 will be needed for the proof of the polycylic case (see Theorem 2.10).
2.3 Finitely generated nilpotent groups
To prove algebraic recurrence of finitely generated nilpotent groups, we will need the following lemma.
Lemma 2.8. LetSbe a subsemigroup of a torsion-free nilpotent groupN, and letS¯be the projection ofStoN/[N, N ]. ThenS = N if and only ifS = N/[N, N ]¯ .
Proof. LetSbe a subsemigroup ofNthat projects to all ofN/[N, N ]. Let1 = Nr/ Nr−1/
· · · / N1/ N0 = N be the lower central series ofN, and suppose by induction that the
lemma holds forN/Z for every cyclic subgroupZ ofNr−1. For each suchZ,Sprojects
to all of(N/Z)/[(N/Z), (N/Z)], and so by induction it projects to all ofN/Z. So to show
N ⊂ S, it suffices to find a cyclic subgroupZ ⊂ Nr−1such thatZ ⊂ S.
Let Z be an arbitrary cyclic subgroup ofNr−1, an letz be a generator ofZ. There
are a and b in N such that [a, b] = z, and by induction S contains representatives of each coset ofZ0, so there arek1, . . . , k4 such thatc = azk1, d = bzk2, e = a−1zk3 and
f = b−1zk4 are all inS. A simple calculation shows thatcndmenfm= znm+L(n,m), where
L(n, m) = (k1+ k3)n + (k2+ k4)m, anddmcnfmen = z−nm+L(n,m). Lettingnandm be
large, we get thatzkandz`are both inS, for somek > 0and` < 0. Together,zkandz`
generate a cyclic subgroupZ0ofZ which is contained inS.
ECP 18 (2013), paper 28.
Theorem 2.9. Every finitely generated nilpotent groupN is AR.
Proof. By Theorem 2.1, we can assume that N is torsion-free. Given a random walk
(X1, X2, . . .)onN, let( ¯X1, ¯X2, . . .)denote the projection ontoN/[N, N ]. The semigroup
generated by{ ¯Xn,Xn+1¯ , . . .}is the projectionS¯nofSn. By Theorem 2.1, we can assume
N/[N, N ] ∼=Zd, and soN/[N, N ]is AR, soS¯n = N/[N, N ]almost surely. By Lemma 2.8, Sn= N almost surely.
2.4 Polycyclic groups
To prove that finitely generated nilpotent groups are AR, we crucially used the fact that every symmetric random walk is Liouville. This is unknown for polycyclic groups. However one has
Theorem 2.10. Every Liouville symmetric non-degenerate random walk on a virtually polycyclic group is AR.
Proof. Let us start with an easy lemma
Lemma 2.11. To prove that a groupGis AR, it suffices to show that any finite index subgroup is AR. Moreover, the same holds if one restricts to Liouville random walks.
Proof. IfH is a subgroup ofG, then aµ-random walk onGcan be projected to a ran-dom walk onG/H. If this random walk is recurrent, then we can define the harmonic measureµH onH. IfG/H isµ-recurrent and(H, µH)is AR, then(G, µ)is AR. Indeed,
the intersection of aµ-random walk onGwithH is aµH-random walk onH (which is
Liouville if the latter is). Because(H, µH)is AR, we must haveH ⊂ Sn almost surely.
But by the recurrence ofG/H, there is a representative of each coset ofH inSn. Thus,
Gis inSn. As a special case of this fact, we see that ifH is a finite index subgroup ofG
andH is AR, thenGis AR.
Now let us turn to the proof of Theorem 2.10. Up to passing to a finite index sub-group, we can assume thatGis (finitely generated nipotent)-by-abelian. Let (Xn) be
a symmetric non-degenerate Liouville random walk on G and let (Sn) be the
corre-sponding sequence of semigroups. SinceG/[G, G]is AR, it is enough to prove that a.s.
Hn := Sn∩ [G, G] = [G, G]. On the other hand, [G, G] being nilpotent, up to dividing
by the derived subgroup of[G, G](which is normal inG), one can assume that[G, G]is abelian: this indeed follows from Lemma 2.8.
Up to dividing Gby a finite normal subgroup, and applying Theorem 2.1, one can assume that[G, G]is torsion-free, hence isomorphic toZk. ClearlyHnis a Liouville
sym-metric decreasing sequence of random semigroups ofZk, so in order to apply Theorem 2.5, it is enough to show thatHn is non-degenerate: this will end the proof of Theorem
2.10.
Claim:Hna.s. generates[G, G]as a subgroup.
Observe that for every integer m ∈ N, the subgroupNm of m-powers of elements of
[G, G]is normal inGand letπm be the projection ofGtoG/Nm. It follows from
Theo-rem 2.1 that a.s.πm(Sn) = G/Nm, and soπm(Hn) = [G, G]/Nm. Recall that every proper
subgroup ofZksits inside the kernel of some surjective morphism
Zk → Z/mZ. In
par-ticular such subgroup does not surject toG/Nmfor the correspondingm. Put together,
Algebraically recurrent random walks on groups
3
The free group
An example of a group that is not AR is the free group on more than four generators. Theorem 3.1. LetFdbe the free group ondgenerators, and letµdbe uniform over the
standard generating set ofFd. Ford > 4,(Fd, µd)is not AR.
LetXn(i, j)denote the symbolsithroughjofXn, written in its reduced form. Letlog
denote log base 2, and letexpdenote exponentiation base 2. We will use the following lemma.
Lemma 3.2. There exists aj0such that 2ith positive probability, for allj > j0there are
at mostlog jstrings of lengthjthat appear as prefixes of someXn.
Proof. Consider a random walk (Y1, Y2, . . .) on Z+, reflected at 0, with probabilities
(2d−1)/(2d)and1/(2d)to increase by one or decrease by one, respectively. The number of strings of lengthjthat appear as prefixes of someXnis upper bounded by the number
Vjof visits tojin this biased random walk. TheVj forj 6= 0are i.i.d. geometric random
variables with parameter p = probability of returning to j. The return probability p
satisfies the equation p = 1/(2d) + 2d1 2d−1
2d p, implying p = 2d
(2d)2−2d+1. We have
Pr(Vj > log j) = plog j, so the probability that there is aj > j0withVj > log jis at most
X
j>j0
plog j = X
j>j0
jlog p< 1,
for sufficiently largej0, sincep < 1/2and thereforelog p < 1ford > 4.
Define:
Ar= {w : |w| = r, w = Xn1(i1, ji) · · · Xnm(im, jm), i1= 1, ik ≤ log jk−1for all2 ≤ k ≤ m}.
We will show that there exists ann0such that with constant probability, all words inSn0
have length-rprefixes inAr.
Lemma 3.3. |Ar| ≤ 4rwith positive probability.
Proof. Let`k = jk− ik. There are2r−1 ways to choose the`k, so we only need to show
that for any fixed{`k}, there are fewer than2rways to choose the{ik}and{Xnk}. We
haveik ≤ log jk−1= log (ik−1+ `k−1), so the number of ways to choose theikis at most
log j1· · · logjm
≤ log `1· log (`2+ log `1) · log (`3+ log (`2+ log `1)) · · · log (`m+ log (`m1+ · · · + log log · · · log `1))
≤Π1≤k≤m(log `k+ log log `k−1+ · · · + log · · · log `1)
= exp
X
1≤k≤m
log(log `k+ log log `k−1+ · · · + log · · · log `1)
≤ exp X 1≤k≤m
(log log `k+ log log log `k−1+ · · · + log · · · log `1)
≤ exp X 1≤k≤m
(log log `k+ log log log `k+ · · · + log · · · log `k+ 1)
≤ exp 2 X 1≤k≤m log log `k ≤ exp(2(r/4)) =2r/2. ECP 18 (2013), paper 28. Page 6/8 ecp.ejpecp.org
where the last inequality follows becauseP
1≤k≤mlog log `k is maximized when all of
the`k are equal to 4. Similarly, Lemma 3.2 tells us that with positive probability, the
number of possible choices of the{Xnk}is also bounded bylog j1· · · log jm≤ 2
r/2. Thus,
the we have|Ar| < 4r, as claimed.
Let A = ∪rAr and d > 4. For reduced words xand y, let cancel(x, y) denote the
number of symbols that are cancelled in the multiplicationx · y.
Lemma 3.4. There exists an n0 such that with positive probability, cancel(Xn, w) <
log |Xn|for anyn > n0and anyw ∈ A.
Proof. For a randomX with|X| = s and a fixedw, we havePr(cancel(X, w) > log s) ≤ (2d − 1)− log s (times a factor 2d−1
2d that we will ignore). For anyw ∈ Ar withr > log n,
the lengthlog n prefix of w is in Alog n. So the probability that there exists a w ∈ A
such that cancel(X, w) > log s is at most |Alog s|(2d − 1)− log s ≤ s− log((2d−1)/4) by a
union bound. When d > 4, we have − log((2d − 1)/4) < −1. Union bounding this over the at mostlog schoices ofXn for which|Xn| = s, and then over alls ≥ s0, we get
P
s≥s0(log s)s
− log((2d−1)/4). Because the sum converges, we can chooses
0large enough
that this sum is strictly less than one. By the transitivity of the random walk, there exists ann0so that with positive probability,|Xn| > sminfor alln > n0.
Proof of Theorem 3.1. The proof is by induction. Xn ∈ A by definition for alln. By
Lemma 3.4, with positive probability, if Xn1· · · Xnm ∈ A for all n1, . . . nm ≥ n0, then
Xn1· · · Xnm+1 ∈ Afor alln1, . . . nm+1≥ n0. Thus,Sn0 ⊂ A 6= Fd.
4
Open Questions
This first study leaves several natural questions open.
• Is(G, µ)-AR a group property? We do not know if it is possible for there to be two symmetric measuresµ1and µ2on a groupGwithhµ1i = hµ2i = Gsuch that
(G, µ1)is AR and(G, µ2)is not AR. A first step towards determining whether this
is possible could be to prove that there is noµfor which(Fd, µ)is AR.
• Is AR preserved under taking finite index subgroups? It follows from Lemma 2.11 that ifGcontains a finite-index AR subgroup, then it is AR. We do not know if the converse holds.
• Non-AR groups. We strongly suspect that the free group on two generators in not AR, but our proof technique is not strong enough to show this, and it does not follow immediately from the fact thatFd is a finite index subgroup ofF2(see the
previous question). On the other hand, perhaps the proof of Theorem 3.1 extends to small cancellation groups with growth at least10n. More generally, it would be
interesting to prove that small cancellation groups or hyperbolic groups are not AR. (There are nonamenable torsion groups [5], ruling out the possibility that no nonamenable groups are AR.)
• What is the structure of the sequence of semigroups in AR groups? For non-AR groups, we know thatSnis not all ofG, but it would be interesting to determine
other properties ofSn. For example, what is the growth ofSn? IsSn transient? Is
the intersection∩nSnempty almost surely? In particular what do the limit sets of
such semigroups look like in free groups?
• Are Liouville random walks algebraically recurrent? Recall that the converse is false (see Section 2.1).
• Infinitely generated groups. Infinitely generated groups present a separate chal-lenge. For example, we do not know if the abelian group⊕ZZis AR.
Algebraically recurrent random walks on groups
• Quantitative versions. It follows from Theorem 3.2 in [4] that if R denotes the range of Brownian motion inRn then a.s. R + R + ... + R dn/2etimes covers
Rn.
A simpler second moment argument gives a similar statement forZd or nilpotent
groups. This suggests studying quantitative variants of algebraic recurrence; for example, when(G, µ)is not AR, estimate the probabilitySn= G.
References
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[3] V. Kaimanovich and A. Vershik, Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3) (1983), 457–490. MR-0704539
[4] D. Khoshnevisan, Y. Xiao and Y. Zhong, Local times of additive Levy processes. Stochastic Process. Appl. 104 (2003), 193–216. MR-1961619
[5] A. Yu. Olshanskii. On the question of the existence of an invariant mean on a group. (Russian) Uspekhi Mat. Nauk 35 4(214) (1980), 199–200. MR-0586204
Acknowledgments. Thank you to Elon Lindenstrauss and Ron Peled for pointing out errors in earlier versions of the proofs of Theorems 2.5 and 3.1, respectively. Thank you also to Uri Bader, Vadim Kaimanovich and Yehuda Shalom for useful comments.
ECP 18 (2013), paper 28.
